Glitchy Z-order in Mesh
The Perspective Transform is having trouble with the minimum Z-clip being set to 0. Go to the N Properties Region of the 3D view > 'View' panel, and set the 'Start' and 'End' of the clip to sensible values for your project, avoiding a Start of 0.

If you want the TL;DR, check out the 'Remapping the Z coordinate' paragraph on this page. Note that, if n=0, then f*n is always 0, and information is lost.